n 



Genesis of the Law of Error. 157 



correlation by Pearson and other mathematical statisticians * 

 — all these and many other applications of the law to physics 

 and social science, no longer based on reasoning from 

 Probabilities, would fall in ruins. 



With reference to Laplace, may be noticed a statement 

 which, though not seriously misleading, adds to the perplexity 

 into which the student of Probabilities is sometimes thrown 

 by Professor Sampson's treatment o£ the subject. He 

 attributes to Laplace the following argument : — " Laplace 

 first offered a demonstration that the error-function possessed 

 a definite form and might be derived from the combination 

 of an unlimited number of small errors individually following 

 any arbitrary laws of distribution " (Congress, p. 166 

 Phil. Mag. p. 349). But, as above pointed out (12), it is 

 remarkable, and has been remarked by a leading authority,, 

 that Laplace did not make this use of his theory, but applied 

 it only to averages of observations. Was the demonstration 

 attributed to Laplace first employed by Morgan Crofton in 

 the Transactions of the. Royal Society, 1870 f ? 



Professor Sampson's reference to Poincare (Congress, 

 p. 168) was calculated, on a first hearing at least, to confirm 

 the impression that justice was not being done to the law of 

 error. It seemed as if an advocate referred to the terms of 

 a statute as in favour of his case, omitting mention of a clause 

 which afforded strong support to the case against him (L3). 

 It may be observed that the Poincare proof is free from the 

 objection which Professor Sampson makes against other 

 proofs, including Morgan Crofton's, which " begin by formu- 

 lating the existence of an error-function " (Phil. Mag. p. 348) J. 



That objection might be formidable if we confined the 

 theory to errors proper. We could not a priori assert that 



* See Yule, ' Outlines of Statistics ' ; Pearson, ' Mathematical Theory 

 of Evolution,' Royal Society passim, and Biometrika. 



f The argument is well stated bv Glaisher in the ' Memoirs of the 

 Astronomical Society,' cited above, dated 1872. 



X Laplace's proof is not open to this objection. The observations 

 whose averages he proves to be obedient to the law of error might be 

 like the " causes of error " (the contributory elements whose aggregate 

 make up an error of observation) according- to Morgan Orofton (Phil. 

 Trans, p. 180 (1870)) such that " a function or curve does not assist our 

 conceptions, and we shall do better to consider the points or 'lots them- 

 selves." (Consider the version of Laplace's proof, Caaib. Phil. Trans. 

 p. 51 (1005). The expression <£(.r) employed for the locus of the frequency 

 need not be a continuous differential)! e function. ) And if the method 

 is employed (though Laplace himself did not so employ it) to deduce the 

 form of the frequency-function pertaining to errors of observation from 

 the fact that an error is approximately the sum of numerous contributory 

 elements, it is not necessary to postulate that the sought function is 

 continuous and different iabie. 



